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We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

In this paper, we prove that the 3-rank of the ideal class group of the imaginary quadratic field Q(√>) is at least 3 for every positive integer n.

This corrigendum describes a number of calculation and typographical errors, primarily in the examples, in the 2017 JNT paper Some Real Quadratic Number Fields with their Hilbert 2-Class Field Having Cyclic 2-Class Group, by Elliot Benjamin.

Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem., 15 pages, to appear in Math. Proc. Cambridge Philos. Soc

We introduce the notion of the Weyl group of a number ring. After that, we give a matrix approach to the study of the ideal class group of a number field, the Bass exact sequence of K-theory of the extension functor, and the capitulation kernel of an extension of number fields.

We describe the Hilbert ideal of the vector invariants of a p-Sylow subgroup of the general linear group.

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

Let L be a number field containing the pth primitive root of unity ζ p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L. Let F be a number field and 𝒪 F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K 2𝒪 F and that of some direct summands of the ideal class group of F(ζ p ).

We analyze the complexity of the computation of the class group structure, regulator, and a system of fundamental units of an order in a certain class of number fields. Our approach differs from Buchmann’s, who proved a complexity bound under the generalized Riemann hypothesis of L ( 1 / 2 , O ( 1 ) ) L(1/2,O(1)) when the discriminant tends to infinity with fixed degree. We achieve a heuristic subexponential complexity in O ( L ( 1 / 3 , O ( 1 ) ) ) O(L(1/3,O(1))) under the generalized Riemann hypothesis when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thomé in the context of algebraic curves over finite fields. We also address rigorously the problem of the precision of the computation of the regulator.

We provide a characterization of integers of the form described in the title in the context of class field theory. A more down-to-earth characterization is given when the extended class number is 1, 2, or 3. For completeness we present a formula for the extended class number. Some examples are also given.

Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to $0$, then the intersection of the rings in $X$ is an integral domain with quotient field $F$ such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to $0$, and (b) one-dimensional Pr\"ufer domains with nonzero Jacobson radical and quotient field $F$. The necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to $0$ is motivated by settings in which the valuation rings considered all dominate a given local ring., Comment: 27 pages; to appear in J. Algebra

Let g be a principal modulus with rational Fourier coefficients for a discrete subgroup of SL2(R) lying in between Γ(N) and Γ0(N)† for a positive integer N. Let K be an imaginary quadratic field. We introduce a relatively simple proof, without using Shimuraʼs canonical model, of the fact that the singular value of g generates the ray class field modulo N or the ring class field of the order of conductor N over K. Further, we construct a primitive generator of the ray class field Kc of arbitrary modulus c over K from Hasseʼs two generators.

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory., 11 pages

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970's as a generalization of Dirichlet's class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant., Comment: 17 pages

In this paper, we focus on a special class of ideal projectors. With the aid of algebraic geometry, we prove that for this special class of ideal projectors, there exist “good” error formulas as defined by C. de Boor. Furthermore, we completely analyze the properties of the interpolation conditions matched by this special class of ideal projectors, and show that the ranges of this special class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.

Following what is basically Kummer’s relatively neglected approach to nonunique factorization, we determine the structure of the irreducible factorizations of an element n n in the ring of integers of a number field K K . Consequently, we give a combinatorial expression for the number of irreducible factorizations of n n in the ring. When K K is quadratic, we show in certain cases how quadratic forms can be used to explicitly produce all irreducible factorizations of n n .

The concept of ideals play an important role in Hilbert Algebras in BCK-algebras, so here we will give some relationships between ideal, fuzzy ideal, commutative fuzzy ideal and weak ideal.

In [1] it gives an ideal of Hilbert Algebras in BCK-algebras, as well as some propositions. So in this paper, we will give a fuzzy Ideal of Hilbert Algebras in BCK-algebras and it’s propositions.

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of Mazur and Wiles (Invent Math 76:179–330, 1984) up to the 2-component, and proves Conjecture 0.1 in Kurihara (J Reine Angew Math 561:39–86, 2003). We also study Brumer’s conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.

Let n(≥ 3) be an odd integer. Let k:= $\Q(\sqrt{4-3^n})\)$ be the imaginary quadratic field and k′:= $\Q(\sqrt{-3(4-3^n)})\)$ the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

Let q be a power of a prime number p. Let k=Fq(t) be the rational function field with constant field Fq. Let K=k(α) be an Artin–Schreier extension of k. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of K. Using these results, we study the p-part of the ideal class group of the integral closure of Fq[t] in K. We also give an analogue of the Rédei–Reichardt formula for K.

We introduce the idele class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito’s class field theory for projective curves (Saito in J Number Theory 21:44–80, 1985).

Text By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do .

The main theorem of this article is an extension of the generalized principal ideal theorem for ideals in Noetherian rings. Instead of requiring the rings to be Noetherian, some natural requirements are imposed on the chains of prime ideals under consideration. The standard (Noetherian) version of the generalized principal ideal theorem is deduced as a corollary and two other applications are presented.

Let K be an imaginary quadratic field, and let F be an abelian extension of K, containing the Hilbert class field of K. We fix a rational prime p > 2 which does not divide the number of roots of unity in the Hilbert class field of K. Also, we assume that the prime p does not divide the order of the Galois group G:=Gal(F/K). Let AF be the ideal class group of F, and EF be the group of global units of F. The purpose of this paper is to study the Galois module structures of AF and EF.

In this paper we give an example of imaginary quadratic number field k such that every ideal of k becomes principal in some proper subfields of the Hilbert class field of k.

Under Greenberg's conjecture, we give an efficient method to compute the p-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the p-part of the ideal class group of the maximal real subfield of Q(√-f, ζ p n+1) in the range 1 < f < 200 and 5 ≤ p < 100000. In order to explain our method, we show an example whose ideal class group is not cyclic.

A common generalization of different notions of ideals met in literature together with the associated notions of distributivity in ordered sets are studied. A restricted prime ideal theorem is proved. Moreover, a new characterization of algebraic topped intersection structures is presented.

In this article we associate to every lattice ideal I L , ρ ⊂ K [ x 1 , … , x m ] a cone σ and a simplicial complex Δ σ with vertices the minimal generators of the Stanley–Reisner ideal of σ . We assign a simplicial subcomplex Δ σ ( F ) of Δ σ to every polynomial F. If F 1 , … , F s generate I L , ρ or they generate rad ( I L , ρ ) up to radical, then ⋃ i = 1 s Δ σ ( F i ) is a spanning subcomplex of Δ σ . This result provides a lower bound for the minimal number of generators of I L , ρ which improves the generalized Krull's principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the A-homogeneous arithmetical rank of a lattice ideal. Finally, we show by a family of examples that the given bounds are sharp.

In his paper (Invent. Math. 109 (1992) 329–350), Solomon finds an information on the prime factorization of an element coming from a circular unit 1 - ζ over the ideal class group of a real abelian number field L, where ζ denotes a root of unity. Using this he obtains an annihilator of the p-Sylow subgroup of the subgroup of the ideal class group of L generated by the classes of prime ideals lying above p. We generalize this result to the circular distributions which has the axiomatic definition of Euler systems as its defining property.

New sufficient conditions for the validity of local class field theory for Henselian valued fields are established. An example is presented to show that these conditions are less restrictive than the applicability of the Neukirch abstract class field theory.

The aim of this paper is to study the relationship between reduction numbers and Borel-fixed ideals in all characteristics. By definition, Borel-fixed ideals are closed under certain specializations which is similar to the strong stability. We will estimate the number of monomials which can be specialized to a given monomial. As a consequence, we obtain a combinatorial version of the well-known Eakin-Sathaye's theorem which bounds the reduction number in terms of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's theorem is attained by the reduction number of a lex-segment monomial ideal. This result answers a question of Conca in the affirmative. We will also show that the reduction number of the lex-segment ideal is bounded exponentially by the reduction number of the given ideal., 11 pages

Several new results on the nonexistence of generalized bent functions are proved by using properties of the decomposition law of primes and the class group of imaginary Abelian number fields.

We prove that, under the assumption of the Generalized Riemann Hypothesis, the exponent of the ideal class group of a CM-field goes to infinity with its absolute discriminant. This gives a positive answer to a question raised by Louboutin and Okazaki [4].

Since class numbers of CM number fields of a given degree go to infinity with the absolute values of their discriminants, it is reasonable to ask whether the same conclusion still holds true for the exponents of their ideal class groups. We prove that under the assumption of the Generalized Riemann Hypothesis this is indeed the case.

Let G be a cyclic group of order p 2 and V be a faithful indecomposable representation of G over a field F of characteristic p. We show that the Hilbert ideal of the invariant ring is generated by polynomials of degree at most | G | whenever dim ⁡ V ≤ 4 p or dim ⁡ V ≥ p 2 − 2 p , proving a conjecture of Derksen and Kemper in this particular case.

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field , are contained in the complex field . Let be the set of all prime numbers. For any algebraic number field , let denote the ideal class group of and, writing for the maximal real subfield of , let denote the kernel of the norm map from to the ideal class group of ; for each , let denote the -class group of , that is, the -primary component of , and let denote the -primary component of . Furthermore, for each , we denote by the ring of -adic integers.

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, but also on the values of the ideal class Dedekind zeta functions. Along the way, we derive an estimate of the number of elements in a given ideal with a certain algebraic norm within a finite hypercube. We provide several examples which measure the accuracy and predictive ability of our theorems., 24 pages. Extends our earlier arxiv submission arxiv.1303.3475

We shall consider the maximal cyclotomic extension of a totally real finite algebraic number field and its ideal class group. We shall investigate the structure of the ideal class group with the action of the cyclotomic Galois group.

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

It is shown that certain classes of Bezout domains have stable range 1, and thus are elementary divisor rings. Included is a strengthening of Roquette's principal ideal theorem which states that the holomorphy ring of a family S of valuation rings of a field K, with S having bounded residue fields, is Bezout. A counterpart is also given where a bound is placed on the ramification indices instead of the residue fields, and these results are applied to rings of integer-valued rational functions over these rings. Along the way, characterizations are given of Prufer domains with torsion class group, Bezout domains, and Bezout domains with stable range 1 in terms of a family { B (t) | t∈K} of numerical semigroups associated with the ring R, and a related family { D (t) | t∈K} of numerical semigroups.